Continuous time random walks in periodic systems: fluid limit and fractional differential equations on the circle

In this paper, the continuous time random walk on the circle is studied. We derive the corresponding generalized master equation and discuss the effects of topology, especially important when Levy flights are allowed. Then, we work out the fluid limit equation, formulated in terms of the periodic version of the fractional Riemann–Liouville operators, for which we provide explicit expressions. Finally, we compute the propagator in some simple cases. The analysis presented herein should be relevant when investigating anomalous transport phenomena in systems with periodic dimensions.

[1]  Vickie E. Lynch,et al.  Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model , 2001 .

[2]  P. Diamond,et al.  The dynamics of marginality and self-organized criticality as a paradigm for turbulent transport , 1996 .

[3]  B. Carreras,et al.  Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .

[5]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[6]  I. Podlubny.  Fractional differential equations , 1998 .

[7]  J. Bouchaud,et al.  Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications , 1990 .

[8]  Melvin Lax,et al.  Stochastic Transport in a Disordered Solid. I. Theory , 1973 .

[9]  K. Chung,et al.  Limit Distributions for Sums of Independent Random Variables. , 1955 .

[10]  V. M. Kenkre,et al.  Generalized master equations for continuous-time random walks , 1973 .

[11]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[12]  Field,et al.  Superconducting vortex avalanches. , 1995, Physical review letters.

[13]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[14]  Drossel,et al.  Self-organized critical forest-fire model. , 1992, Physical review letters.

[15]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[16]  K. Knight.  Stable Non-Gaussian Random Processes Gennady Samorodnitsky and Murad S. Taqqu Chapman and Hall, 1994 , 1997, Econometric Theory.

[17]  Vickie E. Lynch,et al.  A model realization of self‐organized criticality for plasma confinement , 1996 .

[18]  B. Carreras,et al.  The foundations of diffusion revisited , 2005 .

[19]  B. Carreras,et al.  Probabilistic transport models for plasma transport in the presence of critical thresholds: Beyond the diffusive paradigma) , 2005 .

[20]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[21]  R. J. Hastie,et al.  High mode number stability of an axisymmetric toroidal plasma , 1979, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[22]  E. Lu,et al.  Avalanches and the Distribution of Solar Flares , 1991 .

[23]  Shigeo Maruyama,et al.  A molecular dynamics simulation of heat conduction in finite length SWNTs , 2002 .

[24]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[25]  Raul Sanchez,et al.  Probabilistic finite-size transport models for fusion: Anomalous transport and scaling laws , 2004 .

[26]  Joseph W. Haus,et al.  Diffusion in regular and disordered lattices , 1987 .

[27]  E. Montroll,et al.  Random Walks on Lattices. II , 1965 .

[28]  Vickie E. Lynch,et al.  Fractional diffusion in plasma turbulence , 2004 .

[29]  Mixed SOC diffusive dynamics as a paradigm for transport in fusion devices , 2001 .